Completeness type properties of semitopological groups , and the theorems of Montgomery and Ellis

In this paper the class of fan-complete spaces is introduced. Every Gδ-subspace of a pseudocompact space is fan-complete. We prove that a paratopological group is a topological group provided it contains a dense fan-complete subspace. Moreover, if a semitopological group contains a dense Čech-complete subspace, then it is a Čech-complete topological group. This improves some results of A. Bouziad, P. Kenderov, I.S. Kortezov, and W.B. Moors. Some other new results are obtained (see, in particular, Theorems 5.1 and 5.3, and Corollaries 5.4, 6.5 and 6.6).

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